Network of Flexible Capacitive Strain Gauges for Reconstruction of Surface Strain
Jingzhe Wu a,*, Chunhui Song a, Hussam S. Saleem a, Austin Downey a, Simon Laflamme a,b
a
Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA 50011, U.S.A
b Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, U.S.A.
Abstract
Monitoring of surface strain on mesosurfaces is a difficult task, often impeded by the lack of
scalability of conventional sensing solutions. A solution is to deploy large networks of flexible
strain gauges, a type of large area electronics. The authors have recently proposed a soft
elastomeric capacitor (SEC) as an economical skin-type solution for large-scale deployment onto
mesosurfaces. The sensing principle is based on a measurable change in the sensor’s capacitance
upon strain. In this paper, we study the performance of the sensor at reconstructing surface strain
map and deflection shapes. A particular feature of the sensor is that it measures surface strain
additively, because it is not utilized within a Wheatstone bridge configuration. An algorithm is
proposed to decompose the additive in-plane strain measurements from the SEC into principal
components. The algorithm consists of assuming a polynomial shape function, and deriving strain
based on Kirchhoff plate theory. A least-squares estimator (LSE) is used to minimize the error
between the assumed model and the SEC signals after the enforcement of boundary conditions.
Numerical simulations are conducted on a symmetric rectangular cantilever thin plate under
symmetric and asymmetric static loads to demonstrate the accuracy and real-time applicability of
the algorithm. The performance of the algorithm is further examined on an asymmetric cantilever
laminated thin plate constituted with orthotropic materials mimicking a wind turbine blade, and
subjected to a non-stationary wind load. Results from simulations show good performance of the
algorithm at reconstructing the surface strain maps for both in-plan principal strain components,
and that it can be applied in real time. However, its performance can be improved by strengthening
assumptions on boundary conditions. The algorithm exhibits robustness in performance with
respect to load and noise in signals, except when most of the sensors’ signals are close to zero due
to over-fitting form the LSE.
Keywords: Surface strain, deflection shape, shape reconstruction, soft elastomeric capacitor,
structural health monitoring, large area electronics, skin sensor.
1. Introduction
Condition assessment of geometrically large systems, termed mesosystems, which include
transportation infrastructures, aerospace structures, and energy systems, is a difficult task due to
the large surfaces that require monitoring. The majority of existing sensing solutions have
technical and/or economic obstacles that limit their scalability, thus impeding their applicability
[1, 2]. Because it is clear, in some applications, that condition assessment has strong potential
benefits, there is a growing interest in developing sensing solutions that are deployable at large
scales. In particular, it has been thoroughly discussed that monitoring of wind turbine blades may
lead to important reductions in the cost of wind turbine operation and maintenance, which would
decrease costs associated with wind energy production [3-5].
To enable monitoring of mesosystems, the authors have recently proposed a large flexible soft
elastomeric capacitor (SEC) [2, 6]. The sensing principle is based on a measurable change in the
sensor’s capacitance upon strain. Analogous to biological skin, several SECs can be arranged in a
network configuration to measure discrete surface strain over large surfaces, as demonstrated in
Ref. [7] for uni-axial strian. Other skin-type sensors have been proposed including large sensing
sheets of strain gauges [8, 9], resistance-based thin-films fabricated by leveraging the high
conductivity of carbon nanotubes [10-13] and capacitance-based sensors for strain [14], pressure
[15], triaxial force [16], and humidity [17, 18] measurements. The SEC investigated herein differs
from literature by combining high scalability, low-cost fabrication, and mechanical robustness.
The utilization of such dense sensor networks enables the collection of vast arrays of strain data
on mesosurfaces. It is thus possible to conduct condition assessment by reconstructing surface
features, such as strain maps and deflection shapes, and examining temporal changes within
features that can be correlated to altered structural conditions. The reconstruction of strain maps
from displacement data has been proposed using digital image correlation from displacement field
data [19, 20]. Algorithms have also been proposed to reconstruct deflection shapes from strain data
based on the inverse finite element method [21-23], and by directly integrating strain data [24-26].
Others have studied the reconstruction of strain maps directly from strain data using shape
functions either derived from plate theory [27-29] or approximated [30, 31].
The algorithms presented in literature for strain map and deflection shape reconstruction from
strain data require knowledge of the strain magnitude and direction. Unlike most strain sensing
methods, the SEC developed by the authors measures the additive in-plane strain components. It
follows that both the principal strain magnitudes and directions are hidden in the information. To
fully enable the applications of SECs for condition assessment of mesosurfaces, an algorithm must
be developed to retrieve the hidden information used in feature reconstruction.
Here, we propose to leverage the network configuration in the SEC application to decompose
the additive strain measurements and reconstruct strain maps and deflection shapes. The objective
is to enable real-time acquisition of these condition assessment features from SEC data. The
algorithm consists of assuming a shape deformation consistent with boundary conditions, and
deriving two-dimensional strain functions. An LSE is used to obtain the best-fit result for the strain
functions based on the measured additive strain values from the SEC. However, the formulation
of the LSE from the additive sensor’s signal creates multi-collinearity. It results that the proposed
algorithm relies on the quality of the assumptions made about the boundary condition on strain.
The performance of the algorithm is verified on a symmetric thin plate subjected to symmetric and
asymmetric static loads. Also, potential application on wind turbine blades is demonstrated by
simulating an asymmetric cantilever laminated thin plate with orthotropic material resembling a
wind turbine blade.
The paper is organized as follows. Section 2 derives the sensing principle and explains the
additive strain sensing feature. Section 3 presents the strain decomposition algorithm, and validates
the algorithm on a small scale laboratory experiment. Section 4 verifies the algorithm on a simple
cantilever structure subjected to symmetric and asymmetric loads. Section 5 extends the
simulations to an asymmetric laminated plate with orthotropic material properties mimicking a
wind turbine blade. Section 6 concludes the paper.
2. SEC for Surface Strain Monitoring
The SEC technology is described in details in Ref. [6]. Briefly, a soft nanocomposite constitutes
its dielectric, onto which a conductive paint is applied to create the electrodes. Fig. 1(a) shows the
picture of an SEC (only one electrode is shown, in black). Its schematic representation is provided
in Fig. 1(b). The sensor has been designed to be adhered onto surfaces using off-the-shelf epoxies,
along the x-y plane (Fig. 1(b)). Strains in the x or y direction provoke a change in the sensor’s
geometry, which can be measured via a change in the sensor’s capacitance.
(a)
(b)
Fig. 1. (a) A single SEC (70 x 70 mm2); and (b) schematic representation of the SEC.
At low measurement frequency (< 1 kHz), the SEC can be approximated as a non-lossy
capacitor:
𝐶 = 𝑒0 𝑒𝑟
𝐴
ℎ𝑑
(1)
where 𝐶 is the capacitance, 𝐴 = 𝑤 ∙ 𝑙 is the sensor area of width 𝑤 and length 𝑙, ℎ𝑑 is the thickness
of the dielectric, 𝑒0 = 8.854 pF/m is the vacuum permittivity, and 𝑒𝑟 is the relative permittivity
of the polymer. Assuming small strain, Eq. (1) can be differentiated:
∆𝐶
∆𝑙 ∆𝑤 ∆ℎ
(2)
=( +
− ) = 𝜀𝑥 + 𝜀𝑦 − 𝜀𝑧
C
𝑙
𝑤
ℎ
where 𝜀𝑥 , 𝜀𝑦 and 𝜀𝑧 are the three principal directional strains as shown in Fig. 1(b). Assuming no
external stress along the z direction, one can use Hooke’s Law to obtain an expression for 𝜀𝑧 :
𝜈
(3)
𝜀𝑧 = −
(𝜀 + 𝜀𝑦 )
1−𝜈 𝑥
where 𝜈 is the Poisson’s ratio of the SEC. Substituting Eq. (3) into Eq. (2) gives the following
electromechanical model:
∆𝐶
(4)
= 𝜆(𝜀𝑥 + 𝜀𝑦 )
𝐶
where 𝜆 = 1/(1 − ν) is the gauge factor (𝜆 ≈ 2 for the SEC). Eq. (4) evidences that the sensor
measures additive in-plane strains.
3. Algorithm for Strain Decomposition
The algorithm presented in this section is specialized for application to wind turbine blades. It is
based on thin plate and shell structures theory, where shear deformation is assumed to be
insignificant compared with bending deformations. Such assumption is common in literature in
modeling of wind turbine blades, and has been shown to provide accurate results [32]. Note that
the algorithm can be extended to other types of structures by varying the strain-displacement
relationships derived below (Eqs. (8) and (9)), and by modifying the formulation defining
deflection (Eq. (5)).
3.1 Algorithm formulation
Consider the cantilever plate shown in Fig. 2. The following nth order polynomial function can
be used to approximate its deflection shape:
𝑛
𝑤(𝑥, 𝑦) = ∑ 𝑎𝑖𝑗 𝑥 𝑖 𝑦 𝑗
(5)
𝑖=1,𝑗=0
where w is the deflection, and 𝑎𝑖𝑗 are coefficients. Note that 𝑖 ≠ 0 is used to satisfy the
displacement boundary condition on the clamped edge (𝑤(0, 𝑦) = 0).
Fig. 2. Schematic representation of a cantilever thin plate
The displacement 𝑤𝑘 (𝑥, 𝑦) at any sensor k can be represented in matrix notation:
𝐖 = 𝐇𝐀
(6)
where 𝐖 ∈ ℛ 𝑚×1 is the displacement vector, 𝐇 ∈ ℛ 𝑚×𝑛(𝑛+1) is the location matrix, 𝐀 ∈
ℛ 𝑛(𝑛+1)×1 is the coefficient vector, and m is the number of sensors in the sensor network:
𝐖 = [𝑤1 … 𝑤𝑘 … 𝑤𝑚 ]𝑇
𝐇
𝑥1
=[ ⋮
𝑥𝑚
𝑥1 𝑦1
⋮
𝑥𝑚 𝑦𝑚
⋯ 𝑥1 𝑦1𝑛
⋮
⋮
𝑛
⋯ 𝑥𝑚 𝑦𝑚
𝑥12
⋮
2
𝑥𝑚
𝑥12 𝑦1 ⋯ 𝑥12 𝑦1𝑛
⋮
⋮
⋮
2
2 𝑛
𝑥𝑚 𝑦𝑚 ⋯ 𝑥𝑚 𝑦𝑚
⋯
⋮
⋯
𝑥1𝑛
⋮
𝑛
𝑥𝑚
𝑥1𝑛 𝑦1
⋮
𝑛
𝑥𝑚 𝑦𝑚
⋯
⋮
⋯
𝑥1𝑛 𝑦1𝑛
⋮ ]
𝑛 𝑛
𝑥𝑚 𝑦𝑚
(7)
𝐀 = [𝑎10 … 𝑎𝑖𝑗 … 𝑎𝑛𝑛 ]𝑇
Surface strain functions can be obtained from Eq. (5):
𝛆𝐱 = −
𝑐 𝜕 𝟐𝐖
= 𝐇𝐱 𝐀 𝐱
2 𝜕𝑥 2
(8)
𝛆𝐲 = −
𝑐 𝜕 2𝐖
= 𝐇𝐲 𝐀 𝐲
2 𝜕𝑦 2
(9)
and written in terms of sensors’ signals:
𝐒 = [𝑠1
…
𝑠𝑘
… 𝑠𝑚 ]T = 𝛆𝐱 + 𝛆𝐲 = 𝐇𝐬 𝐀 𝐬
(10)
where, for convenience, the signal 𝑠𝑘 for the kth sensor is taken as:
𝑠𝑘 =
∆𝐶𝑘
= 𝜀𝑥,𝑘 + 𝜀𝑦,𝑘
𝜆𝐶𝑘
(11)
and 𝐒 ∈ ℛ 𝑚×1 is the sensor signal vector, 𝐀 = [𝐀 𝟎 |𝐀 𝐬 ] , 𝐇 = [𝐇𝟎 |𝐇𝐬 ] with 𝐀 𝟎 and 𝐇𝟎
representing terms in Eq. (6) that vanish upon double differentiation. Note that 𝐀 𝐱 and 𝐀 𝐲 are
subsets of 𝐀 𝐬 .
The parameter vector 𝐀 s is estimated using an LSE:
Âs= (𝐇𝐬𝐓 𝐇𝐬 )−𝟏 𝐇𝐬𝐓 𝐒
(12)
where the hat denotes an estimation. In its unaltered form, 𝐇𝐬 is multi-collinear because 𝐇𝐱 and
𝐇𝐲 share multiple columns. Matrix 𝐇𝐬𝐓 𝐇𝐬 is therefore singular and non-invertible. To enable full
rank of 𝐇𝐬𝐓 𝐇𝐬 , boundary conditions on the strain map need to be included within 𝐇𝐬 . This is done
by altering the row of 𝐇𝐬 associated with the 𝑘 𝑡ℎ sensor subjected to a particular boundary
condition. These are discussed in what follows for the specialized case of a cantilever plate.
3.2 Boundary Conditions for Cantilever Plate
The displacement boundary conditions are satisfied through the formulation in Eq. (5) via the
selection of appropriate null coefficients. A similar strategy is used for enforcing the boundary
conditions on rotation using the partial derivatives of Eq. (5). In particular, it is assumed here that
the plate does not displace nor rotate at the root. The enforcement of the assumptions on the
boundary conditions on strain needs to be enforced on the second derivatives of the equation as
discussed in Section 3.1. For instance, for the cantilever plate shown in Fig. 2, we assume the
following boundary conditions:
(1) ɛ𝑦 (0, 𝛼𝑦 ≤ 𝑦 ≤ 𝐿𝑦 − 𝛼𝑦 ) = 0;
(2) ɛ𝑦 (𝛼𝑥 ≤ 𝑥 ≤ 𝐿𝑥 − 𝛼𝑥 , 0) = ɛ𝑦 (𝛼𝑥 ≤ 𝑥 ≤ 𝐿𝑥 − 𝛼𝑥 , 𝐿𝑦 ) = − 𝜈𝑚 ɛ𝑥 by taking 𝜎𝑦 = 0;
(3) ɛ𝑥 (𝐿𝑥 , 𝛼𝑦 ≤ 𝑦 ≤ 𝐿𝑦 − 𝛼𝑦 ) = − 𝜈𝑚 ɛ𝑦 by taking 𝜎𝑥 = 0.
where 𝜈𝑚 is the Poisson’s ratio of the plate, 𝛼𝑥 and 𝛼𝑦 are constants such that 0 ≤ 𝛼𝑥 ≤ 𝐿𝑥 and
0 ≤ 𝛼𝑦 ≤ 𝐿𝑦 to account for different boundary conditions at corners [33, 34].
After enforcing these constraints, Âs can be obtained from Eq. (12), which includes the
estimated parameters Âx and Ây. This allows the decomposition of the strain components using Eqs.
(8) and (9). The deflection shape can also be estimated through Eq. (6) after solving 𝐀 𝟎 using the
boundary conditions on displacement and rotation at the clamped edge.
3.3 Experimental Validation
The proposed algorithm is validated on a small scale laboratory experiment. The inexpensive offthe-shelf capacitance data acquisition system (ACAM PCap02) used for the experiment only
measures up to eight capacitors (including a reference), which limits the number of SECs used in
the experimental procedure. In this experiment, six SECs are deployed on a cantilever fiber glass
plate of dimensions (457 x 324 x 3.2 mm3). Resistive strain gauges (RGS) (TML Strain Gauges
type FLA-6-11-3LT) are installed along each edge of each SEC to verify measurements. Data from
the RGS are acquired using a Hewlett-Packard 2850 data acquisition system and sampled at 1.7
Hz, while data from the SECs are sampled at 20 Hz. Two load configurations are considered: 1) a
point load at the center tip; and 2) a point load at a corner. These two load configurations are
consistent with loads used in the numerical simulations (Section 4). They were applied sequentially
while taking continuous measurements. Fig. 3 shows the laboratory test configuration. Data from
the SECs are compared against the average value of RSG measurements located on all four sides
of each sensor.
(a)
(b)
Fig. 3. Laboratory test configuration: (a) picture; and (b) schematic. Distances are in mm.
Figure 3 shows the strain values acquired from each SEC using Eq. (4). They are compared against
the theoretical reading from the summing readings from surrounding RSGs. The time series counts
five plateaus. The first three plateaus are used to calibrate the sensor (accurately establishing C0).
The fourth plateau show results from the application of the center load, while the fifth plateau
show results from the application of the corner load. Time series results show a good fit between
the SECs and the RSGs approximately within the reported resolution of 25𝜇𝜀 in [6], validating the
electromechanical model.
Fig. 4. Strain values obtained from SEC’s plotted against RSG results
Given the low number of sensors used in this laboratory verification, the boundary conditions can
only be enforced in one or two sensors in order to demonstrate the algorithm. In addition, because
the monitored plate itself is small compared with possible full-scale applications, the sensors are
relatively far from the edge which adds error in the boundary condition assumptions. For both load
cases, the only assumed boundary condition is 𝜀𝑦 = 0 at the fixed edge (thus for sensors C and F).
The average values at each plateau for each SEC are taken as input for the algorithm. The
decomposition is conducted using a deflection shape (Eq. (5)) of the fourth order in x and third
order in y given the low number of sensors used on the plate.
Tables 1 and 2 show the decomposition results compared with the average results acquired form
the RSGs located on both sides of each sensors in each directions, for the center and corner load
cases, respectively. Results show an overall acceptable performance for both load cases for
computing principal strain component values, except for sensor D that showed a substantial
overestimation. Factors that contribute to larger discrepancies include 1) the signal-to-noise ratio
in the SEC signal, which is notably observable in Fig. 4 for sensor D; 2) the weak assumption on
the boundary condition for sensors C and F, in particular for the center load case; and 3) the load
number of sensors used in the experiment, which limits the application of a consistent polynomial
(Eq. (5)) and additional boundary conditions. Nevertheless, this laboratory experiment
demonstrates the promise of the algorithm at decomposing principal strain components.
Table 1. Center load (units are in 𝜇𝜀).
sensor
A
B
C
D
E
F
𝜺𝒙
𝜺̂𝒙
|𝜺𝒙 − 𝜺̂𝒙 |
𝜺𝒚
𝜺̂𝒙
|𝜺𝒚 − 𝜺̂𝒚 |
186.9
366.8
574.4
202.1
383.8
572.3
177.8
330.0
559.9
411.3
423.1
512.6
9.02
36.8
14.4
209.1
39.21
59.77
-54.78
-33.10
-24.85
-49.51
-24.72
-20.27
-77.82
5.93
0.00
-252.5
-82.18
0.00
23.04
39.03
24.85
203.0
57.46
20.27
Table 2. Corner load (units are in 𝜇𝜀).
sensor
A
B
C
D
E
F
𝜺𝒙
𝜺̂𝒙
|𝜺𝒙 − 𝜺̂𝒙 |
𝜺𝒚
𝜺̂𝒙
|𝜺𝒚 − 𝜺̂𝒚 |
136.0
341.7
534.1
165.5
374.0
589.8
185.5
339.8
542.8
385.4
465.0
593.3
49.50
1.90
8.69
219.9
91.01
3.48
38.18
9.74
0.65
37.12
19.15
-5.56
-11.62
10.89
0.00
-178.1
-77.22
0.00
49.80
1.14
0.65
215.2
96.37
5.56
4. Simulations – Symmetric Plate
This section verifies the performance of the strain decomposition algorithm on an isotropic
rectangular cantilever thin plate under different static loads. The influence of the order of the
polynomial defining the deformation (Eq. (5)) and the robustness of the algorithm under different
noise levels are examined as well as computing time.
4.1 Numerical model
The simulated model consists of a cantilever aluminum thin plate, with SECs deployed in a
network configuration. Two load cases are studied: load case 1 is a point load of 2 kN at the center
tip to produce a symmetric response; and load case 2 is a 1.5 kN load at a corner to produce torsion.
Each SEC is assumed to have a size of 70 x 70 mm2, similar to a typical size (Fig. 1(a)). Fig. 5
shows the numerical model. Each square in Fig. 5 denotes an SEC, the filled circles denote the
centroid of a sensor, red circles denote a sensor subjected to boundary conditions (discussed in
Sect. 3.2), and blue circles denote sensors not subjected to boundary conditions. A 2% Gaussian
noise is introduced in the simulated SEC signals (constructed with Eq. (11)); the surface strains in
two principal directions are reconstructed using Eqs. (8) and (9); and the deflection shapes are
reconstructed using Eq. (6). The algorithm is verified with the highest polynomial order possible
with the selected sensor arrangement (𝑛 = 6). The absolute percentage error (APE) and mean
absolute percentage error (MAPE) are used as performance measures for the algorithm. The APE
is used to compare fitting results over a contour map, while the MAPE is used to compare the
overall fitting results over the entire plate.
Fig. 5. Simulated symmetric plate model. Distances are in mm.
4.2 Strain decomposition
(a)
(b)
(c)
(d)
Fig. 6. APE contour within plate boundary: (a) APE-ɛ𝑥 under load case 1; (b) APE-ɛ𝑦 under load case 1; (c)
APE-ɛ𝑥 under load case 2; (d) APE-ɛ𝑦 under load case 2.
Fig. 6 compares the APE of the strain decomposition results for strains ɛ𝑥 and ɛ𝑦 in the two
principal directions of the cantilever plate under load cases 1 and 2. Results are shown for the
SECs not subjected to boundary conditions (blue circles in Fig. 5). The algorithm performs well
at estimating surface strain along the x-axis, which dominates the bending behavior. This fitting
performance is lower at the free corners and at the load location. This can be caused by corner
singularities and local shear stress caused by point load which introduce more error in the
assumptions on the strain boundary conditions and plane stress assumption for thin plate. Also, the
boundary conditions are applied at the modeled boundary sensor nodes which are not located
directly on the boundaries. Fig. 6 (b) and (d) shows a higher APE for the estimation of the surface
strain along y-axis under both load cases. This can be caused by a higher dependence of ɛ𝑦 on the
boundary condition assumptions, which are less accurate at the free-end corners. In addition, the
APE of ɛ𝑦 is more significant under torsion (load case 2) at localized region on the clamped edge.
The three-dimensional (3-D) maps of strain decomposition results for load case 1 and 2 are
shown in Fig. 7. While the APE suggests regions of larger errors, the 3-D strain maps display a
good agreement in the overall shape, with the largest errors located at the plate boundaries. This
shows the high dependence of the algorithm on the assumptions made on the boundary conditions.
Fig. 8 shows the deflections shapes reconstructed using Eq. (6) for both load cases. There is good
agreement between the analytical and estimated results. The MAPE of deflection shapes is 4.76%
for load case 1 and 2.62% for load case 2. The error around the point load location can also be of
particular interest given the possible error introduced by the inapplicability of the thin plate inplane stress assumption. The APE is 3.10% for load case 1 and 3.31% for load case 2 directly at
the point load location, and the MAPE at the neighbor sensor nodes is 2.60% and 1.57% for load
cases 1 and 2, respectively. The assumption of thin plate deformation provided a good estimation
for the reconstruction of strain maps.
(a)
(b)
(c)
(d)
Fig. 7. 3-D strain map over full plate surface: (a) ɛ𝑥 under load case 1; (b) ɛ𝑦 under load case 1; (c) ɛ𝑥 under
load case 2; (d) ɛ𝑦 under load case 2.
(a)
(b)
Fig. 8. (a) Deflection shape under load case 1; (b) Deflection shape under load case 2.
4.3 Polynomial order and noise
(a)
(b)
(c)
(d)
Fig. 9. Variation of MAPE versus polynomial order nd noise level: (a) MAPE-ɛ𝑥 under load case 1; (b)
MAPE-ɛ𝑦 under load case 1; (c) MAPE-ɛ𝑥 under load case 2; (d) MAPE-ɛ𝑦 under load case 2.
This performance of the algorithm with respect to the polynomial order of the deflection shape
(Eq. (5)) and signal noise is examined. Fig. 9 shows the MAPE of the estimation results for all the
non-boundary sensors versus the polynomial order and noise level for load case 1 and 2. The order
of the polynomial varies from a second order to a sixth order, the highest possible order given the
number and configuration of the SEC network (Fig. 5). The investigated noise levels are 0%, 2%,
5%, 10%, 15% and 20% Gaussian noise. Results show that at least a fourth order polynomial is
required to provide an adequate performance. Increasing the polynomial order beyond this point
provides a marginal increase in performance for most cases. Also, the algorithm appears to be
stable under noise levels below 10%.
Table 3. Computing time as a function of the polynomial order.
order level
computing time (ms)
2nd
7.89
3rd
8.41
4th
8.58
5th
8.74
6th
9.02
The influence of the polynomial order on the computation time is shown in Table 3. Simulations
were conducted in MATLAB using a 2.50 GHz Intel CPU. Increasing the order of the polynomial
does not significantly increase computing time. The longest computing time of 9.02 ms enables a
real-time application for a typical sampling frequency below 100 Hz. Note that this performance
can be improved by using different software, code, and computing platform.
5. Simulations – Asymmetric plate
Fig. 10. Simulated asymmetric plate model. Dimensions are in mm.
(a)
(b)
Fig. 11 Time histories of simulated wind loads: (a) wind pressure 1; (b) wind pressure 2.
5.1 Numerical model
Numerical simulations are extended to an asymmetric plate mimicking a wind turbine blade.
The 9 meters cantilever thin plate is constituted with laminated sections of orthotropic materials
as defined in Ref. [35]. The model is subjected to two time varying wind pressures on two
separated areas. The simulated asymmetric plate model along with the configuration of the SEC
network is shown in Fig. 10. Similarly to Fig. 5, the red circles denote the sensors subjected to
boundary conditions, and the blue circles denote the sensors not subjected to boundary conditions.
Two sets of non-stationary wind pressure time series were produced at 5 Hz over 60 seconds using
a dual approach based on low frequency measured real wind speed data and turbulence spectrum
[36, 37]. Time histories are shown in Fig. 11.
5.2 Strain decomposition
(a)
(b)
Fig. 12. MAPE of time histories: (a) ɛ𝑥 ; (b) ɛ𝑦 .
Fig. 12 shows the time series of the MAPE for the sensors not subjected to boundary conditions
(blue circles in Fig. 10). During most of the wind excitation, the MAPE of both ɛ𝑥 and ɛ𝑦 remains
approximately constant and low: around 3% for ɛ𝑥 and 12% for ɛ𝑦 . The fit on strain ɛ𝑦 has a much
higher variability in results, as it would be expected from the simulation results discussed in the
previous section. Nevertheless, the generally constant error and its relatively low level demonstrate
stability of the algorithm with respect to a time-varying excitation. The higher value peaks in the
MAPE coincide with sensor signals close to 0 for the majority of sensors, which cause over-fitting
from the LSE algorithm. A possible alternative would be to apply a different fitting algorithm
when several sensors output measurements close to 0. This is out-of-the-scope of this paper.
Fig. 13 shows the APE contours for a typical simulation result when the MAPE is stable (taken
at 30 sec). Unlike for the symmetric plate case, both the APE contours of the estimation of ɛ𝑥 and
ɛ𝑦 suggest a relatively higher error at the fixed root. This can be attributed to the error in the
assumptions on the boundary conditions caused by an irregular geometry. The APE on most
regions of the plate remains under 5% for ɛ𝑥 and under 15% for ɛ𝑦 . Fig. 13 (c) is a 3-D plot of the
deflection shape, showing good agreement between the estimated and analytical results. The
MAPE for the deflection shape is 2.70 % and the maximum APE is 12.14 % over the entire surface.
(a)
(b)
(c)
Fig. 13. Strain decomposition results at 30 sec: (a) APE-ɛ𝑥 ; (b) APE-ɛ𝑦 ; (c) deflection shape.
6. Conclusions
We have presented a novel sensor for surface strain measurements. Used in a network
configuration, SECs can be used to cover mesosurfaces, and measurements used to reconstruct
physics-based features for condition assessment, such as strain maps and deflection shapes. Given
that each SEC measures additive strain from both principal directions, an algorithm was developed
to retrieve the magnitude and directional information of strain prior to reconstructing strain maps.
The algorithm assumes a polynomial displacement shape, and an LSE is used to estimate the
coefficients of the polynomial after enforcement of the boundary conditions. Given the proposed
engineering application to wind turbine blades, the algorithm was specialized for thin plates and
shell structures. The performance of the algorithm has been verified on a symmetric cantilever thin
plate with SECs arranged in a network configuration. Further investigations have been conducted
on an asymmetric cantilever laminated thin plate with orthotropic materials mimicking a wind
turbine blade, subjected to a non-stationary wind load. Results showed a good robustness and
accuracy of the algorithm under most loading conditions, except when most of the sensors’ signals
are close to zero.
While results demonstrate the overall promise of the algorithm, refining the assumptions on the
boundary conditions and developing strategies to cope with corner singularities have the potential
to substantially improve its accuracy. Future work could include a dual form that would eliminate
over-fitting from the LSE when the sensors’ signals are close to zero.
We have showed that the sensor network can be used to produce two-dimensional strain maps
and deflection shapes, in real-time and for noise levels under 10%. The study of temporal and
spatial changes in these maps and shapes could be used for real-time condition assessment of
mesosystems. This would significantly empower system managers and owners with conducting
timely maintenance and optimizing operations.
Acknowledgements
This work is supported from the Iowa Alliance for Wind Innovation and Novel Development
(IAWIND) (Grant 1001062565), and the Iowa Energy Center (Grant 13-02). Their support is
gratefully acknowledged.
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